This document is a project report submitted as part of an undergraduate electrical engineering curriculum. It investigates solving a problem involving the mutual inductance of a coupled circuit. The objective is to determine and plot the voltage v1(t) and v2(t) across the inductors. Theoretical calculations are shown using equations relating the voltages and currents based on the self and mutual inductances. A MATLAB simulation is also presented, showing the results match the theoretical calculations.

1. Project Report
On
PROBLEM SOLVING ON THE MUTUAL INDUCTANCE OF COUPLED CIRCUIT
Submitted as a part of Second Year First Semester curriculum of
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL AND ELECTRONICS ENGINEERING
By
Mr. P.Mohan 18981A0238
Ms. P.Supriya 18981A0239
Mr. P. Nithesh Kumar 18981A0240
Mr. P.Sanjay Kumar 18981A0241
Mr.P.Bharat 18981A0242
Mr.P.Venkatesh 18981A0243
Under the Supervision of
Dr.G.vasu
Associate Professor
Department of Electrical & Electronics Engineering
Raghu Engineering College
(Autonomous)
Accredited by NBA & NAAC with ‘A Grade, Permanently Affiliated JNTU
Kakinada
Dakamarri (v), Bheemunipatnam Mandal, Visakhapatnam, Andhra Pradesh
531162
Dec – 2019

2. RAGHU ENGINEERING COLLEGE
Department of Electrical and Electronics
Engineering
CERTIFICATE
This is to certify that case study entitled as “PROBLEM SOLVING ON THE
MUTUAL INDUCTANCE OF COUPLED CIRCUIT” submitted by Mr.P.Mohan
(18981A0238), Ms.P.Supriya(18981A0239), Mr. P.NitheshKumar(18981A0240), Mr.P.Sanjay
Kumar(18981A0241) Mr.P.Bharat (18981A0242)Mr.P.Venkates(18981A0243)
as a part of Second Year First Semester Curriculum of Under Graduate
Program in Electrical & Electronics Engineering at Raghu Engineering
College.
Faculty Member Head of the department
Dr.G.vasu Dr P. Sasi Kiran
Associate Professor Professor

3. Objective :-
The objective is that we have to solve the given problem by using the different solving techniques.
From this
a.) Determine and plot for v1(t).
b.) Determine and plot for v2(t).
Principles and methodologies :-
There are three popular circuit analysis methods. All three produce the same answer.
 Direct application of the fundamental laws (Ohm's Law and Kirchhoff's Laws)
 Node Voltage Method
 Mesh Current Method and its close relative, the Loop Current Method
The first method, direct application of the fundamental laws, is quick and works very well for
simple circuits. It is not particularly efficient in terms of the total amount of work required, which
becomes important as circuits become more complicated.
Engineers have come up with two elegant ways to organize and streamline circuit analysis: the
Node Voltage Method and the Mesh Current Method. These are general-purpose step-by-step
recipes to solve a circuit. Both methods attempt to minimize the number of simultaneous
equations. This efficiency has a big impact as circuit complexity grows (more and more nodes and
branches). The Loop Current Method is a close relative of the Mesh method, used in certain
special cases, as described in that article.
As we study the methods of circuit analysis, our example circuits are made of only resistors and
ideal sources. This keeps the math relatively simple, allowing us to concentrate on the strategies
for solving a circuit . Calculate the mutual inductance M

4. Necessity of solving the given objective.
However, when the emf is induced into an adjacent coil situated within the same magnetic field,
the emf is said to be induced magnetically, inductively or by Mutual induction, symbol ( M ). Then
when two or more coils are magnetically linked together by a common magnetic flux they are said
to have the property of Mutual Inductance.
Mutual Inductance is the basic operating principal of the transformer, motors, generators and any
other electrical component that interacts with another magnetic field. Then we can define mutual
induction as the current flowing in one coil that induces a voltage in an adjacent coil.
But mutual inductance can also be a bad thing as “stray” or “leakage” inductance from a coil can
interfere with the operation of another adjacent component by means of electromagnetic induction,
so some form of electrical screening to a ground potential may be required.
The amount of mutual inductance that links one coil to another depends very much on the relative
positioning of the two coils. If one coil is positioned next to the other coil so that their physical
distance apart is small, then nearly all of the magnetic flux generated by the first coil will interact
with the coil turns of the second coil inducing a relatively large emf and therefore producing a large
mutual inductance value.
The important parameters is
 L1 and L2 represent inductances.
 M represents a mutual inductance with coefficient of coupling K.
 v1(t) and v2(t) represent the voltage across the terminals of the inductors at time t.
 i1(t) and i2(t) represent the current through the inductors at time t. The block uses standard dot notation to
indicate the direction of positive current flow relative to a positive voltage.
Theoretical calculations of the objective.
According to the given problem
The given data is
Mutual inductance (m) = 2uH
Self-inductance (L1) = 4mH
Self-inductance (L2) = 10mH
According to the given problem

5. The mutual induced voltage equation is
V1 = L1
𝑑𝑖1
𝑑𝑡
+ 𝑀
𝑑𝑖2
𝑑𝑡
V2 = L2
𝑑𝑖2
𝑑𝑡
− 𝑀
𝑑𝑖1
𝑑𝑡
Since here the i2 is open circuit
i2 = 0
so the equations will become
V1 = L1
𝑑𝑖1
𝑑𝑡
V2 = −𝑀
𝑑𝑖1
𝑑𝑡
by taking the time intervals as shown in the current graph
i1 = 2t for 0 < t < 1
i1 = -t+3 for 1< t < 4
i1 = t-5 for 4< t < 5
so by solving the equation we can get by equation 1
here by taking different instantaneous times we can get
V1(t) = 4 * 10-3 *
𝑑(2𝑡)
𝑑𝑡
for 0 < t < 1
V1(t) = 4 * 10-3 *
𝑑(−𝑡+3)
𝑑𝑡
for 1< t < 4
V1(t) = 4 * 10-3 *
𝑑(𝑡−5)
𝑑𝑡
for 4< t < 5
Finally
V1(t) = 8 * 10-3 for 0 < t < 1
V1(t) = -4 * 10-3 for 1< t < 4
V1(t) = 4 * 10-3 for 4< t < 5
The graph is shown below
1
2

6. This values are in the micro voltage
Now calculating the value of v2
V2 = L2
𝑑𝑖2
𝑑𝑡
− 𝑀
𝑑𝑖1
𝑑𝑡
V2 = −𝑀
𝑑𝑖1
𝑑𝑡
So by using the equation 2
We can get
by taking the time intervals as shown in the current graph
i1 = 2t for 0 < t < 1
i1 = -t+3 for 1< t < 4
i1 = t-5 for 4< t < 5
V2 = -2 * 10-3 *
𝑑(2𝑡)
𝑑𝑡
V2 = -2 * 10-3 *
𝑑(−𝑡+3)
𝑑𝑡
V2 = -2 * 10-3 *
𝑑(𝑡−5)
𝑑𝑡
By doing derivative
V2 = -4 * 10-3 for 0 < t < 1
V2 = 2 * 10-3 for 1< t < 4
V2 = -2 * 10-3 for 4< t < 5
-6
-4
-2
0
2
4
6
8
10
0 1 2 3 4 5 6
V1 Values
0
1
1
4
4
4
5
5

7. The graph is shown below
-5
-4
-3
-2
-1
0
1
2
3
0 1 2 3 4 5 6
V2 Values
0
1
1
4
4
5
5

8. Simulation of the problem
This is the simulation of the program that we have solved the program in the matlab.
The matlab Simulink for the given circuit and the simulation results.
Here that is the code of the given coupled circuit.
And there is also a given simulation results of the circuit.

9. MATLAB CODE
clear;
t=0:1:5;
for i=1:6
if i<=1
I(i)=2*t(i);
else if i>1 & i<=4
I(i)=-t(i)+3;
else
I(i)=t(i)-5;
end
end
end
%figure(1);
p1=plot(t,I*1D-3);
hold on
PriV1=4D-3*2;
PriV2=4D-3*-1;
PriV3=4D-3*1;
V1_1=PriV1.*ones(1,length(t));
V1_2=PriV2.*ones(1,length(t));
V1_3=PriV3.*ones(1,length(t));
SecV1=-2D-3*2;
SecV2=-2D-3*1;
SecV3=-2D-3*1;
V2_1=SecV1.*ones(1,length(t));
V2_2=SecV2.*ones(1,length(t));
V2_3=SecV3.*ones(1,length(t));
figure(1)
title('I vs V1 vs V2')
Current_plot=plot(t,I*1D-3,'b');
hold on
plot(t(1:2),V1_1(1:2),'k');
hold on
plot(t(2:5),V1_2(1:4),'k');
hold on
p2=plot(t(5:6),V1_3(5:6),'k');
hold on
plot(t(1:2),V2_1(1:2),'r');
hold on
plot(t(2:5),V2_2(1:4),'r');
hold on
p3=plot(t(5:6),V2_3(5:6),'r');
ylim([PriV2-2D-3 PriV1+2D-3]);
legend([p1 p2 p3],{'Current','Primary Voltage','Secondary
Voltage'});
xlabel('Time');
figure(2)
Current_plot=plot(t,I*1D-3,'b');
title('Current vs Time');
xlabel('Time');

10. ylabel('Current');
figure(3)
plot(t(1:2),V1_1(1:2),'k');
hold on
plot(t(2:5),V1_2(1:4),'k');
hold on
p2=plot(t(5:6),V1_3(5:6),'k');
title('Primary Voltage vs Time')
ylim([PriV2-2D-3 PriV1+2D-3]);
legend([p1 p2 p3],{'Current','Primary Voltage','Secondary
Voltage'});
xlabel('Time');
ylabel('Voltage');
hold on
figure(4)
plot(t(1:2),V2_1(1:2),'r');
hold on
plot(t(2:5),V2_2(1:4),'r');
hold on
p3=plot(t(5:6),V2_3(5:6),'r');
title('Secondary Voltage vs Time');
ylim([PriV2-2D-3 PriV1+2D-3]);
legend([p1 p2 p3],{'Current','Primary Voltage','Secondary
Voltage'});
xlabel('Time');
ylabel('Voltage');
Comparison of theoretical and simulation results
From the above both results shows the similar characteristics for the coupled circuits .And by that
results and graphs are similar to each other in the circuit
Figure of plotting the current used in time the current from this we can conclude that both the
therotical and the graphical representation shows the same results

11. This figure is plot between the voltage and the time graph the current from this we can conclude
that both the therotical and the graphical representation shows the same results
Plottings of the v1(t) and the current from this we can conclude that both the therotical and the
graphical representation shows the same results

12. Plottings of the v2(t) the current from this we can conclude that both the therotical and the
graphical representation shows the same results
And here in this problem the independent current source is present from that we can solve both
the circuits .and here one side is the current is open circuit so we can the solve problem.
Conclusion
From this the problem we can understand that above results are equal.
There is no much difference between the both the output results and simulation results are both
are equal.
Lm is the mutual inductance, such that Lm ≤ GL1⋅L2
The Mutual Inductance block can be used to model two- or three-windings inductances with equal
mutual coupling, or to model a generalized multi-windings mutual inductance with balanced or
unbalanced mutual coupling.